In 2015, the price of a business math text rose to $150. This is 8% more than the 2014 price. What was the old selling price?

Final answer:

The general solution to the given second-order differential equation, 3y'' + 2y' + y = 0, is found using the characteristic equation method, resulting in complex roots. The solution is expressed in terms of sine and cosine functions multiplied by an exponential decay factor.

Explanation:

To find the general solution of the given second-order differential equation, 3y'' + 2y' + y = 0, we first convert it into its characteristic equation. This is done by substituting y = ert into the differential equation, where r is the root of the characteristic equation and t is an independent variable. This approach transforms the given differential equation into a quadratic equation.

The characteristic equation for this differential equation is 3r2 + 2r + 1 = 0. Solving this quadratic equation using the formula r = [-b ± sqrt(b2 - 4ac)] / 2a, where a=3, b=2, and c=1, gives the roots of the characteristic equation. In this case, the discriminant (b2 - 4ac) is less than zero, indicating complex roots.

The roots can be found to be r = -1/3 ± i(sqrt(2)/3). Therefore, the general solution to the differential equation is y(t) = e-t/3[C1cos(sqrt(2)t/3) + C2sin(sqrt(2)t/3)], where C1 and C2 are constants determined by initial conditions.

The general solution is [tex]\( y(t) = e^{-\frac{t}{3}} (C_1 \cos\left(\frac{\sqrt{2} t}{3}\right) + C_2 \sin\left(\frac{\sqrt{2} t}{3}\right)) \)[/tex].

To solve the second-order linear homogeneous differential equation [tex]\( 3y'' + 2y' + y = 0 \)[/tex], we follow these steps:

1. Write the characteristic equation associated with the differential equation.

2. Solve the characteristic equation for its roots.

3. Write the general solution based on the roots of the characteristic equation.

Step 1: Write the Characteristic Equation

The given differential equation is:

[tex]\[ 3y'' + 2y' + y = 0 \][/tex]

We assume a solution of the form [tex]\( y = e^{rt} \)[/tex]. Substituting [tex]\( y = e^{rt} \)[/tex] into the differential equation, we get:

[tex]\[ 3(r^2 e^{rt}) + 2(r e^{rt}) + e^{rt} = 0 \][/tex]

Dividing through by [tex]\( e^{rt} \)[/tex] (which is never zero), we obtain the characteristic equation:

[tex]\[ 3r^2 + 2r + 1 = 0 \][/tex]

Step 2: Solve the Characteristic Equation

The characteristic equation is a quadratic equation:

[tex]\[ 3r^2 + 2r + 1 = 0 \][/tex]

To find the roots of this quadratic equation, we use the quadratic formula:

[tex]\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

where [tex]\( a = 3 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 1 \)[/tex].

Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:

[tex]\[ r = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \][/tex]

[tex]\[ r = \frac{-2 \pm \sqrt{4 - 12}}{6} \][/tex]

[tex]\[ r = \frac{-2 \pm \sqrt{-8}}{6} \][/tex]

[tex]\[ r = \frac{-2 \pm 2i\sqrt{2}}{6} \][/tex]

[tex]\[ r = \frac{-1 \pm i\sqrt{2}}{3} \][/tex]

Thus, the roots of the characteristic equation are:

[tex]\[ r_1 = \frac{-1 + i\sqrt{2}}{3} \][/tex]

[tex]\[ r_2 = \frac{-1 - i\sqrt{2}}{3} \][/tex]

Step 3: Write the General Solution

Since the roots are complex conjugates [tex]\( r_1 = \alpha + i\beta \)[/tex] and [tex]\( r_2 = \alpha - i\beta \)[/tex] with [tex]\( \alpha = -\frac{1}{3} \)[/tex] and [tex]\( \beta = \frac{\sqrt{2}}{3} \)[/tex], the general solution to the differential equation is of the form:

[tex]\[ y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)) \][/tex]

Substitute [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:

[tex]\[ y(t) = e^{-\frac{t}{3}} \left( C_1 \cos\left( \frac{\sqrt{2} t}{3} \right) + C_2 \sin\left( \frac{\sqrt{2} t}{3} \right) \right) \][/tex]

Thus, the general solution of the differential equation [tex]\( 3y'' + 2y' + y = 0 \)[/tex] is:

[tex]\[ y(t) = e^{-\frac{t}{3}} \left( C_1 \cos\left( \frac{\sqrt{2} t}{3} \right) + C_2 \sin\left( \frac{\sqrt{2} t}{3} \right) \right) \][/tex]

where [tex]\( C_1 \)[/tex] and [tex]\( C_2 \)[/tex] are arbitrary constants.

Answer:

the answer is add 2 feet on each side!

Step-by-step explanation:

The equivalent fractions with the same denominator are 6/ 15 and 5 /15.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the two fractions are 2/5 and 1/3. The two equivalent expressions with the common base will be written as:-

2/5 = ( 2 x 3 ) / ( 5 x 3 )

2//5 = 6 / 15

1/3 = ( 1 x 5 ) / ( 3 x 5 )

1 / 3 = 5 / 15

Therefore, the two fractions are 6/ 15 and 5 /15.

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The expression that shows identity property of multiplication is (x+yi)×1=(x+yi), the correct option is D.

According to the identity property of multiplication, the product of 1 and a number is the number itself.

The expressions in the question are

(X+yi)×z=(xz+yzi)

(x+yi)×0=0

(x+yi)×(z+wi)=(z+wi)×(x+yi)

(x+yi)×1=(x+yi)

The only expression that shows this property is (x+yi)×1=(x+yi).

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The derivative of the function f'(x)= arccsc 8x is f'(x)= -1/(|8x|√((8x)² - 1)). We achieved this result by using the rules for derivatives of arcsine, arcsecant and arccosecant along with the chain rule for differentiation of composite functions.

To find the derivative of the function f'(x)= arccsc 8x, we will first need to understand that the derivative of the arcsine of x, also known as the inverse sine of x, is 1/(√(1 - x²)). Similarly, the derivative of arcsecant of x, known as the inverse secant of x, is 1/(|x|√(x² - 1)).

However, here we have arccosecant of x, known as the inverse cosecant of x. With this, we find that the derivative of arccosecant of x is -1/(|x|√(x² - 1)). Now in context of our function, where we replace x with 8x, we get f'(x)= -1/(|8x|√((8x)² - 1)).

We used the rules mentioned as well as the mutation rule ( df (u) = [du f(u)] dx ). This mutation rule is also known as the chain rule in differentiation, which allows us to differentiate composite functions. The function present here is a composite function where we have 8x in place of x in the arccosecant function.

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The answer is:

5 hours, 246 miles.

Since we know the equations that represent the distances of both cars at certain times (function of time), we can calculate the time that they will be at the same distance by making their equation equal.

So, we are given the equations:

A-

[tex]y=44x+26[/tex]

B -

[tex]y=36x+66[/tex]

Now, by making both equation equal, we can calculate the time that they will have the same distance, so, we have:

[tex]44x+26=36x+66[/tex]

[tex]44x-36x=66-26[/tex]

[tex]8x=40[/tex]

[tex]x=\frac{40}{8}=5[/tex]

Hence, we have that they will have the same position after 5 hours.

Then, to calculate the distance, we need to substitute the obtained time in any of the given equations, so, substituting into A, we have:

[tex]y=44x+26[/tex]

[tex]y=44*5+26=220+26=246[/tex]

We have that the distance will be 246 miles.

Hence, we have that the answer is:

5 hours, 246 miles.

Have a nice day!

Answer:  The total number of fifth graders in the school is 155.

Step-by-step explanation:  Given that at Brown Elementary School, 80% of all fifth graders ride the bus to school.

If 124 fifth graders ride the bus to school, we are to find the number of fifth graders in the the school.

Let, 'n' be the total number of fifth graders in the school.

Then, according to the given information, we have

[tex]80\%\times n=124\\\\\\\Rightarrow \dfrac{80}{100}\times n=124\\\\\\\Rightarrow \dfrac{4}{5}n=124\\\\\\\Rightarrow n=\dfrac{124\times 5}{4}\\\\\\\Rightarrow n=31\times5\\\\\Rightarrow n=155.[/tex]

Thus, the total number of fifth graders in the school is 155.

It says to check if [tex] \frac{3}{8} [/tex] satisfy each equation or not.

Checking first equation :-

96 x [tex] \frac{3}{8} [/tex]

⇒ [tex] \frac{96 * 3}{8} [/tex]

⇒ [tex] \frac{288}{8} [/tex] = 36

⇒ 36 = 36. First equation is "YES".

Checking second equation :-

38 x [tex] \frac{3}{8} [/tex]

⇒ [tex] \frac{38 * 3}{8} [/tex]

⇒ [tex] \frac{114}{8} [/tex] = [tex] \frac{57}{4} [/tex]

⇒ [tex] \frac{57}{4} [/tex] ≠ 14. Second equation is "NO".

Checking third equation :-

16 x [tex] \frac{3}{8} [/tex]

⇒ [tex] \frac{16 * 3}{8} [/tex]

⇒ [tex] \frac{48}{8} [/tex] = 6

⇒ 6 = 6. Third equation is "YES".

Checking fourth equation :-

56 x [tex] \frac{3}{8} [/tex]

⇒ [tex] \frac{56 * 3}{8} [/tex]

⇒ [tex] \frac{168}{8} [/tex] = 21

⇒ 21 = 21. Fourth equation is "YES".

The value of x is option (D) 2

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.

Given,

f(x) = [tex]2^{x}[/tex]

we have to find the value of x when f(x) =4

f(x) = [tex]2^{x}[/tex] =4

[tex]2^{x}=4\\ 2^{x}=2^{2}[/tex]

Therefore value of x is 2

Hence, the value of x is option (D) 2

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Final answer:

The surface area of a right rectangular prism with dimensions of 6cm by 14cm by 5cm is calculated by summing up the areas of all faces, resulting in a total surface area of 368cm².

Explanation:

To calculate the surface area of a right rectangular prism, we need to find the area of all the faces and add them together. For a prism with dimensions of 6cm by 14cm by 5cm, it has three pairs of identical faces: two faces that are 6cm by 14cm, two faces that are 6cm by 5cm, and two faces that are 14cm by 5cm.

Using the area formula for rectangles (Area = length × width), we find the areas of these faces:

For the 6cm by 14cm faces: Area = 6cm × 14cm = 84cm². Since there are two such faces, their combined area is 2 × 84cm² = 168cm².

For the 6cm by 5cm faces: Area = 6cm × 5cm = 30cm². Since there are two such faces, their combined area is 2 × 30cm² = 60cm².

For the 14cm by 5cm faces: Area = 14cm × 5cm = 70cm². Since there are two such faces, their combined area is 2 × 70cm² = 140cm².

Adding up all these areas gives us the total surface area of the prism:

Total Surface Area = 168cm² + 60cm² + 140cm² = 368cm²

Therefore, the surface area of the right rectangular prism is 368cm².

Answer with explanation:

Question asked by Mr.Maddox to find the sum of fractions with the help of number line :

    [tex]\rightarrow 3 \frac{2}{3}+1 \frac{3}{4}+\frac{2}{3}\\\\ \text{Using Associative Property}\\\\\rightarrow a+(b+c)=(b+c)+a\\\\\rightarrow [\frac{2}{3}+ 1\frac{3}{4}]+3 \frac{2}{3}\\\\\rightarrow [\frac{2}{3}+\frac{7}{4}]+\frac{11}{3}\\\\\rightarrow \frac{21+8}{12}+ \frac{11}{3}\\\\\rightarrow\frac{29+44}{12}\\\\ \rightarrow\frac{73}{12}\\\\\rightarrow 6\frac{1}{12}[/tex]

Answer:

sample response:  None of the roots can have multiplicity because the polynomial is cubic and 3 roots are given. Write each root as a linear factor, then multiply the three factors to get the expression for the function.

Step-by-step explanation:

Answer:

C){(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)}

Step-by-step explanation:

The one-to-one function has inverse where the inverse is also function.

That is, there should be unique output for each input values.

Look at the options, Option C) only has unique output for each input values.

Therefore, the answer C){(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)}

Hope this will helpful.

Thank you.

Answer:

h^2=a^2+b^2.

h^2=(6x^2y)^2+(9x^2y)^2.

h^2=36x^4y^2+81x^4y^2.

h^2=117x^4y^2.

h=sqrt(117x^4y^2).

=3 √13 x^2y

After applying Pythagoras' theorem the length of the hypotenuse we get is approximately 10.8 x²y unit.

Use the concept of the triangle defined as:

A triangle is a three-sided polygon, which has three vertices and three angles which has a sum of 180 degrees.

And the Pythagoras theorem for a right-angled triangle is defined as:

(Hypotenuse)²= (Perpendicular)² + (Base)²

Given that,

Base = 6x²y

perpendicular = 9x²y

Now apply the Pythagorean theorem,

(Hypotenuse)²= (6x²y)² + (9x²y)²

(Hypotenuse)²= 36x⁴y² + 81x⁴y²

(Hypotenuse)²= 117x⁴y²

Take square root on both sides we get,

Hypotenuse = √117 x²y

Hypotenuse ≈ 10.8 x²y

Hence,

The length of the hypotenuse is approximately 10.8 x²y unit.

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We can convert inches to meters with the following conversion formula:

1 inch = 0.0254 meters

Now we have to find how many meters are there in 276 inches.

So multiplying 0.0254 with 276 would give us the answer.

276 inches = 0.0254 *276 = 7.0104 meters.

Answer : There are 7.0104 meters in 276 inches.

we know that

To find out the fraction of a pan of brownies that was left subtract [tex]1\frac{1}{5}[/tex] from [tex]2[/tex]

remember that

[tex]1\frac{1}{5}=(1*5+1)/5=\frac{6}{5}[/tex]

so

[tex]2-\frac{6}{5} =(2*5-6)/5=\frac{4}{5}[/tex]

therefore

the answer is

the fraction of a pan of brownies that was left is [tex]\frac{4}{5}[/tex]

I think its both because An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. Consecutive means one after the other. The constant value is called the common difference. Another way to think about an arithmetic sequence is that each term in the sequence is equal to the previous term plus the common difference and also A geometric sequence is a sequence in which the ratio of any two consecutive terms is constant. This constant value is called the common ratio. Another way to think about a geometric sequence is that each term is equal to the previous term times the common ratio. So when u look at it check out if the 16, 8, 4, and 2 see if they fit these terms

The ratios are constant (0.5), so the sequence is geometric.

Since the sequence is not arithmetic but is geometric, the correct answer is: B. Geometric

To determine whether the sequence 16, 8, 4, 2 is arithmetic, geometric, both, or neither, let's analyze the differences between consecutive terms and the ratios between consecutive terms.

Arithmetic Sequence:

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.

Calculating the differences:

8 - 16 = -8

4 - 8 = -4

2 - 4 = -2

The differences are not constant, so the sequence is not arithmetic.

Geometric Sequence:

A geometric sequence is a sequence in which the ratio between consecutive terms is constant.

Calculating the ratios:

8 / 16 = 0.5

4 / 8 = 0.5

2 / 4 = 0.5

The ratios are constant (0.5), so the sequence is geometric.

Since the sequence is not arithmetic but is geometric, the correct answer is:

B. Geometric

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The given points do not form a rhombus.

A rhombus is a quadrilateral that has four equal sides.

Some of the properties we need to know are:

- The opposite sides are parallel to each other.

- The opposite angles are equal.

- The adjacent angles add up to 180 degrees.

We have,

To determine if the given points form a rhombus, we need to check if the sides are congruent (have equal length) and if the opposite angles are congruent (have equal measure).

First, we can find the lengths of all four sides of the quadrilateral using the distance formula:

AB = √((3 - 0)² + (4 - 0)²) = 5

BC = √((8 - 3)² + (4 - 4)²) = 5

CD = √((5 - 8)² + (0 - 4)²) = 5

DA = √((0 - 5)² + (0 - 4)²) = 5

Since all four sides have the same length of 5 units, the quadrilateral satisfies the property of having congruent sides.

Next, we need to check if the opposite angles are congruent.

We can do this by finding the slopes of the two diagonals and checking if they are perpendicular. If the slopes are perpendicular, then the opposite angles are congruent.

The slope of diagonal AC can be found as:

m(AC) = (4-0)/(8-0) = 1/2

The slope of diagonal BD can be found as:

m(BD) = (0-4)/(5-3) = -2/2 = -1

Since the product of the slopes is:

m(AC) x m(BD) = (1/2) x (-1) = -1/2

which is not equal to -1, the diagonals are not perpendicular and the opposite angles are not congruent.

Therefore,

The given points do not form a rhombus.

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